The Three Little Pigs Arrange Their Houses

Date: 05/13/2002 at 13:33:38
From: D. Boiros
Subject: Word Problem
My daughter got a problem work sheet on the Three Little Pigs and
one of the questions we came up with two different answers for.
The question was: How many different ways could the pigs arrange
their houses?
I come up with 27 by using 3*3*3 = 27. But when we go to list
the different combinations we can only get 6. Could you please
tell us the correct answers?
Thank you.
Mrs. Boiros

Date: 05/13/2002 at 16:13:23
From: Doctor Jeremiah
Subject: Re: Word Problem
Hi there Mrs. Boiros! I am glad you wrote.
I think this is a very interesting question. The answer you
computed is too large because it includes arrangements that
include the same house more than once.
Let A, B, and C represent the houses belonging to the first,
second, and third pigs respectively.
All the possible arrangements are:
A A A A A B A A C A B A A B B A B C A C A A C B A C C
B A A B A B B A C B B A B B B B B C B C A B C B B C C
C A A C A B C A C C B A C B B C B C C C A C C B C C C
But notice that this includes using the same house multiple times
(which obviously really isn't possible with houses).
The reason it comes out to 27 is because the first house can be
one of 3 choices, and the second house can be one of 3 choices,
and the third house can also be one of 3 choices, so the number
of possible arrangements is:
3 (first) x 3 (second) x 3 (third) = 27
Now, like we said before, this isn't the right way to go for
houses because you can't use the same house multiple times, so
the first house is a choice of 3, and then the second house
in the row is a choice of 2 (because one is already used), and
then the third house isn't a choice at all (because the other two
are already used).
So the total number of arrangements (excluding duplicates) would
be:
3 (first) x 2 (second) x 1 (third) = 6
Now we have a lot fewer possible arrangements!
- - - - - - - - - - - - - - - A B C - - - A C B - - -
- - - - - - B A C - - - - - - - - - B C A - - - - - -
- - - C A B - - - C B A - - - - - - - - - - - - - - -
Does that help explain the difference?
Here is a bit more info on this and more complicated topics:
http://www.mathforum.org/dr.math/faq/faq.comb.perm.html
- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/